What is a Commutator in Group Theory?
Definition
For two elements $g, h \in G$ of a group $G$, the commutator of $g, h$ is defined as follows.
$$ [g,h] := ghg^{-1}h^{-1} $$
Explanation
According to the definition, it is equivalent that the commutator of all elements is the identity element $e$ and that $G$ is an abelian group.
The commutator mentioned in group theory and the commutators that appear in quantum mechanics, differential geometry, etc., are slightly different. The commutator in quantum mechanics, differential geometry is the one in ring theory.